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Properties

Label 2-185-185.104-c1-0-4
Degree $2$
Conductor $185$
Sign $0.438 - 0.898i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.160 + 0.0583i)2-s + (0.245 − 0.674i)3-s + (−1.50 + 1.26i)4-s + (−1.21 + 1.87i)5-s + 0.122i·6-s + (3.51 − 0.618i)7-s + (0.338 − 0.586i)8-s + (1.90 + 1.59i)9-s + (0.0856 − 0.371i)10-s + (−2.79 + 4.84i)11-s + (0.483 + 1.32i)12-s + (−1.40 + 1.17i)13-s + (−0.526 + 0.304i)14-s + (0.966 + 1.28i)15-s + (0.664 − 3.76i)16-s + (4.85 + 4.07i)17-s + ⋯
L(s)  = 1  + (−0.113 + 0.0412i)2-s + (0.141 − 0.389i)3-s + (−0.754 + 0.633i)4-s + (−0.544 + 0.838i)5-s + 0.0500i·6-s + (1.32 − 0.233i)7-s + (0.119 − 0.207i)8-s + (0.634 + 0.532i)9-s + (0.0270 − 0.117i)10-s + (−0.843 + 1.46i)11-s + (0.139 + 0.383i)12-s + (−0.388 + 0.325i)13-s + (−0.140 + 0.0812i)14-s + (0.249 + 0.330i)15-s + (0.166 − 0.942i)16-s + (1.17 + 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842084 + 0.526090i\)
\(L(\frac12)\) \(\approx\) \(0.842084 + 0.526090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.21 - 1.87i)T \)
37 \( 1 + (5.12 + 3.27i)T \)
good2 \( 1 + (0.160 - 0.0583i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.245 + 0.674i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-3.51 + 0.618i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.79 - 4.84i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.40 - 1.17i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.85 - 4.07i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.400 + 1.10i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (4.15 + 7.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.90 - 1.67i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.55iT - 31T^{2} \)
41 \( 1 + (0.879 - 0.737i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + (-4.82 + 2.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.99 - 1.58i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (9.96 + 1.75i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.36 - 2.81i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-10.4 + 1.84i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.13 - 2.23i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 6.54iT - 73T^{2} \)
79 \( 1 + (-14.1 + 2.49i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.232 + 0.277i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (11.2 + 1.99i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.67 + 2.90i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.54772178784620001397816179404, −12.13838530283966605928568865636, −10.62627490208768244482655078156, −10.03814040162893840135673745806, −8.276935306704183442773151169355, −7.76273644834393889196588066670, −7.06155946278434563028965382057, −4.92203061171692201084050727205, −4.10973551611748123422603356101, −2.21278522411269344093178834463, 1.06660900810849212382544871926, 3.67186516313831243157739901396, 5.02278255795495103121988502028, 5.49143245659064409164896987371, 7.76985261883751390159076821806, 8.392096294783807696365252321567, 9.407104239667480503321126400910, 10.32831616568344223752347996590, 11.46433632081380507919870398125, 12.32293443320283809195911859429

Graph of the $Z$-function along the critical line